Use of induction logging tools is well-known in the well logging field. (See, e.g., U.S. Pat. Nos. 5,157,605 and 5,041,975, both incorporated herein by reference, and the references cited therein.)
Generally speaking, in "induction logging," one or more transmitter coil(s), energized by alternating current(s), is/are disposed in a borehole (either on a wireline or as part of a logging-while-drilling apparatus) and indication are obtained of the influence of surrounding formations on the electromagnetic field established by the coil(s). Usually, such indications are obtained by observing the voltage(s) induced in one or more receiver coil(s) disposed in the borehole in a coaxial relationship with the transmitter coil(s).
In order to interpret the responses obtained from an induction logging tool (or, for the matter, any sort of downhole tool), it is necessary to understand how the "responds" to various subsurface formations. This mapping between subsurface formation features and tool response is referred to as a "model," and the process of creating a "model" is referred to as "modeling." For an induction logging tool, the critical subsurface parameter is resistivity. Thus, in induction logging, "modeling" involves determining the tools response to a given resistivity distribution in the region of the tool. Such modeling is often referred to as "resistivity modeling."
In the case of purely vertical wells, one can make assumptions about the formations resistivity distribution (e.g., that it is symmetric about the axis of the well) which greatly reduce the complexity of computing the tool response. Unfortunately, such assumptions break down in the increasingly-common realm of deviated and horizontal wells. Such wells demand full, 3D solutions. And, even using the best available techniques and hardware, solving the general, 3D problem is very costly. See B. Anderson et al., "New Dimensions in Modeling Resistivity," Schlumberger Oilfield Review, vol. 9, no. 1, pp. 41-56 (Spring, 1997). Among persons skilled in the art, the need for improved methods of resistivity and tool response modeling is well known.
Accordingly, one object of the invention relates to an improved method, apparatus, and article of manufacture for modeling the response of induction logging and other electromagnetic tools to subsurface formations.
Another object of the invention involves an improved method, apparatus, and article of manufacture for modeling the response of induction logging and other electromagnetic tools using a Spectral Lanczos Decomposition ("SLDM") method.
Yet another object of the invention concerns an improved method, apparatus, and article of manufacture for modeling the response of induction logging and other electromagnetic tools using an SLDM method, with Krylov subspaces generated from the inverse powers of the Maxwell operator.
Still another object of the invention relates to an improved method, apparatus, and article of manufacture for modeling the response of induction logging and other electromagnetic tools using an improved SLDM method, in which convergence is independent of conductivity contrast or frequency.
A still further object of the invention involves interpretation of subsurface formations using modeling data supplied by the aforementioned methods, apparatus, and articles of manufacture.